Radial bound states in the continuum for polarization-invariant nanophotonics

All-dielectric nanophotonics underpinned by the physics of bound states in the continuum (BICs) have demonstrated breakthrough applications in nanoscale light manipulation, frequency conversion and optical sensing. Leading BIC implementations range from isolated nanoantennas with localized electromagnetic fields to symmetry-protected metasurfaces with controllable resonance quality (Q) factors. However, they either require structured light illumination with complex beam-shaping optics or large, fabrication-intense arrays of polarization-sensitive unit cells, hindering tailored nanophotonic applications and on-chip integration. Here, we introduce radial quasi-bound states in the continuum (radial BICs) as a new class of radially distributed electromagnetic modes controlled by structural asymmetry in a ring of dielectric rod pair resonators. The radial BIC platform provides polarization-invariant and tunable high-Q resonances with strongly enhanced near fields in an ultracompact footprint as low as 2 µm2. We demonstrate radial BIC realizations in the visible for sensitive biomolecular detection and enhanced second-harmonic generation from monolayers of transition metal dichalcogenides, opening new perspectives for compact, spectrally selective, and polarization-invariant metadevices for multi-functional light-matter coupling, multiplexed sensing, and high-density on-chip photonics.

The rise of optical metasurfaces has launched a variety of breakthrough applications ranging from negative refraction 1 and ultrathin optical elements 2 to photonic computation 3 . Similarly, bound states in the continuum 4 (BICs) have been shown to underpin many fundamental oscillatory phenomena 5 and can be employed for tailoring the lifetimes of resonant wave systems 6 . Initially discovered in quantum physics 7,8 and also found in acoustics 9,10 and ocean science 11 , BICs have emerged as an intriguing concept in optics [12][13][14] . In all-dielectric subwavelength structures, radiative losses of BIC-based systems can be precisely controlled by tailoring interferences of the constituent resonant modes, producing sharp resonances with extremely high values of the quality (Q) factors. The BIC concept has therefore been employed for a variety of applications driven by such spectrally selective nanosystems, including multiplexed biospectroscopy [15][16][17][18] , high-harmonic generation 19,20 , and subwavelength lasing [21][22][23] .
Two of the principal BIC implementations in nanophotonics are supercavity modes in isolated structures such as individual nanoantennas 24,25 and resonances in symmetry-protected metasurfaces 26 . Although single BIC structures offer a minimal footprint, careful electromagnetic engineering is required to satisfy the demanding mode interference conditions of the hybridization of the Mie and Fabry-Pérot modes 24 , restricting design flexibility and limiting the range of achievable Q factors. Based on these design constraints, such interference-type modes can only be excited with structured light illumination, further increasing experimental complexity. Moreover, the resonantly enhanced electric near fields are predominantly confined inside the structures, which is advantageous for material-intrinsic processes such as higher harmonic generation, but severely limits surface-driven light-matter interactions and sensing applications.
In contrast, metasurfaces based on all-dielectric resonators with broken in-plane inversion symmetry (Fig. 1a) offer both high-Q factors and strong near fields that extend considerably outside of the resonators, enabling cutting-edge applications in sensing 16,18 and enhanced light-matter interaction 20,27,28 . Significantly, symmetry-broken quasi-BICs (qBICs) provide straightforward tunability of the resonance position and Q factor via the geometrical scaling factor of the unit cell 16 and the degree of asymmetry for increased design versatility. However, they require specific excitation polarizations, and the lateral onchip footprints of metasurfaces supporting qBIC resonances in the visible spectrum usually exceed 100 µm 2 (see refs. 18,19), holding back their potential for device miniaturization, on-chip multiplexed sensors, and interactions with hybrid nanophotonic systems such as micronsized individual layers of two-dimensional (2D) materials. Although their spatial extent can be reduced by transforming the system into a one-dimensional chain (Fig. 1b), this comes at a significant cost in terms of Q factor 29 (see Supplementary Note 1).
Here, we introduce the concept of radial quasi-bound states in the continuum (radial BICs) as a multi-application platform for sustaining polarization-invariant high-Q resonances with high surface sensitivity in a compact footprint. Notably, ring structures composed of as few as 12 unit cells can efficiently couple incident light into a pronounced radial quasi-BIC mode while providing flexible resonance tuning and strong surface-confined near fields. In our experimental realization, we achieve high-Q factors exceeding 500 in the red part of the visible spectrum and demonstrate the viability of our approach as a versatile BIC-based platform for biomolecular sensing and enhanced second-harmonic generation in atomically thin 2D semiconductor molybdenum diselenide (MoSe 2 ) monolayers, all in a spatial footprint as small as 2 µm 2 .

Design and implementation of the radial BIC concept
The symmetry-protected and radially distributed electromagnetic BIC states are accessed through a carefully designed ring structure incorporating symmetry-broken double rod unit cells, where individual resonators are rotated to satisfy radial alignment (Fig. 1c). At resonance, adjacent rods support opposing electric dipoles, which leads to a suppression of the total dipole moment and consequently of the radiative losses in the system. In contrast to previous 2D or 1D resonator arrangements, our approach leverages a semi-infinite ring geometry 30,31 , providing a polarization-invariant optical response and avoiding edge effects, which allows a higher number of resonator elements to participate in the qBIC state for increased resonance quality 32 . Owing to the unique circular arrangement, the coupling between elements over the entire ring is improved as well, resulting in much higher Q factors compared to linear resonator chains (Supplementary Fig. S4). As a result, the radial BIC platform provides the highest quality factor to footprint ratio amongst comparable BICbased nanophotonic approaches [18][19][20]32 (Fig. 1). Furthermore, even though neighboring resonators are set at a small angle with respect to each other, the radial arrangement of rods with identical lengths produces a non-radiating BIC state with negligible coupling to the far field (Fig. 2a).
One of the key parameters to control the optical characteristics of the radial BIC mode is the asymmetry parameter ΔL, i.e., the length difference between the two constituent rods of the unit cell. In the symmetric case (ΔL = 0), we observe a BIC-like state, which is decoupled from the radiation continuum and exhibits negligible electromagnetic near-field enhancement (Fig. 2a). For nonzero asymmetry (ΔL > 0), the BIC transforms into a radial quasi-BIC mode (termed radial BIC from now on), which can be accessed from the far field, demonstrating strong field enhancements and the characteristic field pattern of oppositely oriented electric dipole moments observed in asymmetric metasurfaces 26 (Fig. 2b). Due to the engineered inter-resonator coupling within the ring, linearly polarized light efficiently couples to the full radial BIC mode extended along the ring and excites spectrally distinct resonances with high-Q factors (Supplementary Fig. S1). Our numerical calculations confirm the presence of highly enhanced near fields outside the resonator volume and highlight how the field magnitude can be tailored via the structural asymmetry (Fig. 2b). We engineer the radial BIC system to exhibit high-Q resonances at around 770 nm and find that the corresponding numerically simulated Q factors exceed 8000 toward the smallest asymmetries ( Supplementary Fig. S2). Furthermore, we observe a clear correlation between the electric near fields (Fig. 2b, for field intensities, see Supplementary Fig. S6) and the values of the Q factors, where the lowest asymmetry is associated with the maximum values of both parameters.
Symmetry-broken ring structures with varying geometry were fabricated from a 120 nm thick amorphous silicon layer on a glass substrate using electron beam lithography and reactive ion etching (see "Methods" for details). All measurements shown in the manuscript are based on ring structures with 24 annular unit cells and a base rod length L 0 = 335 nm with width w = 115 nm. Symmetry-breaking is introduced by shortening one rod within the unit cell as indicated by the corresponding asymmetry parameter ΔL (Fig. 2b). The choice of the asymmetries is based on tradeoff between highest Q factors (and associated strong near fields) and the resulting modulation of the radial BIC resonances in the optical experiments. Although higher field enhancements and ultrahigh Q factors above 8000 are numerically predicted for asymmetries approaching ΔL = 0 nm (see Supplementary  Optical characterization of the structures is carried out with a commercial confocal microscopy setup under collimated white light illumination. We first examine the optical response of the radial BIC structures with different radii and find precise tuning of the resonance wavelength via the ring radius R (Fig. 2c). Importantly, our versatile design provides additional degrees of freedom for tailoring the resonance, such as the number of unit cells constituting the ring and the base length L 0 of the all-dielectric rods (Supplementary Fig. S8).
Consistent with numerical simulations, we observe the highest resonance sharpness for the smallest values of the asymmetry ΔL with Q factors exceeding 500 (Fig. 2d, for fitting details and full spectra, see Supplementary Figs. S9 and S10). To our knowledge, this is one of the highest Q factors measured for symmetry-broken qBIC resonances in the visible wavelength range. For all asymmetries, the Q factor increases with decreasing radii ( Supplementary Fig. S3), which we attribute to the improved coupling between the resonators. This scaling behavior is consistent with the fact that the coupling vanishes for infinite radii with infinite spacing between the resonators. Strikingly, the annular arrangement of the resonator elements renders the Q factor mostly invariant under rotations of the incident polarization direction of light 33 , maintaining a value above 200 throughout the angular range (Fig. 2e). We attribute the observed spectral polarization invariance to the inherent C 24 symmetry of our ring design which congruently transforms the ring onto itself for a rotation by one unit cell (15°). Furthermore, we do not observe any polarization angle dependence of the spectral response for smaller rotation angles (0 ≤ φ ≤ 15 , see Supplementary Fig. S7). As a result, we conclude that the spectral response of the radial BIC is fully polarization-invariant for any arbitrary polarization angle. Crucially, this polarization-independent performance enables simplified experimental measurements and greatly eases practical device applications.

Enhanced biomolecular sensitivity
We leverage the high-Q resonances and strong surface-confined near fields of the radial BIC platform to demonstrate biomolecular sensing for different ring geometries and asymmetries based on a biotinstreptavidin binding bioassay (Fig. 3a). As an initial step, we assess the refractometric sensing performance by varying the local refractive index around the ring structure (see "Methods" for details) by means of magnetron sputtering of conformal silicon dioxide (SiO 2 ) thin films with increasing thickness. Pronounced resonance shifts for all asymmetries can be detected, depending on the SiO 2 layer thicknesses (Fig. 3b). For the calculation of the bulk refractive index sensitivity S B (BRIS), the resonance shifts mediated by the thin films are converted by considering both the electric near-field decay length of the resonators and the film thickness as introduced previously 34 (see "Methods").
Based on these BRIS values, we find that the sensing figure of merit where FWHM is the resonance full width at half maximum, is clearly correlated with the asymmetry of the radial BIC structures, with the highest FOM of above 20 per refractive index unit (RIU) observed for the lowest asymmetry (Fig. 3c). Despite the much smaller footprint, the FOM values of our radial BIC geometry for biomolecular sensing are mostly comparable with large array-based approaches relying on symmetry-protected metasurfaces 18 . Nevertheless, we believe that the FOM can be substantially boosted by utilizing even smaller asymmetries or by further engineering the inter-rod coupling, e.g., by optimizing the resonator shape. Additional advantages of the radial BIC approach can be realized by a concentric arrangement of several rings for multiplexed operation without increasing the footprint, or by integrating multiple rings with identical resonance wavelengths to enhance light absorption efficiency per area.
For the implementation of a model multistep bioassay, we first functionalize our structures with (3-Aminopropyl) triethoxysilane (APTES), followed by the attachment of biotin molecules and the binding of different concentrations of streptavidin protein. After each step, we observe a clear redshift of the radial BIC resonance (Fig. 3d), induced by the higher number of molecules on the surface and the associated increase of the environmental refractive index. To compare the biomolecular sensing performance for all asymmetries, we normalize the resonance shift by the respective FWHM and plot it for each streptavidin concentration to obtain a map of biomolecular sensing performance (Fig. 3e).
Notably, a clear and streptavidin-dependent sensor response over a broad range of concentrations and resonance linewidths is observed. Depending on the sensitivity and spectral resolution of the spectroscopic equipment in experiments and thus the required resonance modulations as well as Q factors, we demonstrate a wide parameter space for picking the appropriate structural asymmetry. Especially, fielddeployed applications with low spectral resolution may require lower Q resonances-whereas when pushing for highest sensitivity, high-Q resonances and consequently a high FOM is favored. Our versatile radial BIC platform allows to cover all such use cases in an ultrasmall footprint, pushing the limits of on-chip multiplexing applications for biosensing and opening up new avenues for the precise on-demand tailoring of FOM, Q factor, resonance position, and resonance modulation.

Enhanced SHG in an evanescently coupled monolayer of MoSe 2
To further demonstrate the versatility of our design, we employed the radial BIC platform for enhanced light-matter interaction between the all-dielectric resonator system and a two-dimensional excitonic material 20,35,36 . Specifically, we demonstrate and localize enhanced second-harmonic generation (SHG) in a non-centrosymmetric monolayer crystal of the transition metal dichalcogenide (TMD) MoSe 2 , facilitated by the highly enhanced electromagnetic near fields that are associated with the radial BIC resonances. For this purpose, we deterministically transferred a monolayer of MoSe 2 on top of the radial BIC platform (Fig. 4a). Crucially, this experiment is enabled by the low footprint of the radial BIC platform, which allows full spatial overlap between the micron-sized MoSe 2 layer and the ultracompact ring geometry. Atomic force microscopy (AFM) imaging confirms the  homogeneous coverage of the resonators (Fig. 4b) and indicates that the monolayer is in good contact with the silicon structures. In Fig. 4c, we show the quadratic dependence of the integrated SHG signal on the input power for the monolayer together with its polarization-resolved signal (inset, see "Methods"), confirming the origin of the signal from the nonlinear harmonic light generation in the atomically thin crystal 37 .
To demonstrate spectrally selective SHG, we excite the coupled system of a MoSe 2 monolayer on top of a tailored radial BIC structure at three different excitation wavelengths with the use of a tunable femtosecond pulsed laser (Fig. 4d, e, see "Methods" for details). For this purpose, we choose a ring with ΔL = 50 nm to balance the high near-field enhancements with a sufficient resonance modulation. Although we expect even higher near-field and thus SHG enhancements for smaller asymmetries (see Supplementary Fig. S6), these resonances are often strongly damped by the TMD monolayer in experiments. For our SHG measurements, we excite the system at 744 nm in resonance with the radial BIC mode, at 800 nm close to the MoSe 2 exciton peak, and at 900 nm, spectrally distinct from both the exciton and the radial BIC resonance (Fig. 4e). At each excitation wavelength, we raster the focused laser beam across the sample, direct the light to an avalanche single photon detector, and plot the obtained SHG signal intensity for each point. In Fig. 4d, we show the maps of the SHG signal on the ring normalized to the bare MoSe 2 monolayer flake to guarantee consistency between the measurements and to eliminate wavelength-dependent processes such as detector and SHG efficiency as well as transmission of the signal in the optical beam path (see Supplementary Fig. S11b for raw data without background).
For radial BIC-resonant excitation, we observe a fivefold SHG intensity enhancement confined to the ring structure, illustrating the highly localized generation of the second-harmonic signal. In contrast, for off-resonant excitation, we observe a suppression of the SHG (Fig. 4d) within the same region. The suppression of the SHG signal when excited spectrally apart from the radial BIC resonance can be attributed to strain introduced in the monolayer TMD by the ring structure 38 , which is outperformed by the highly enhanced near fields of the resonant excitation. The wavelength dependence is a clear indication that the SHG enhancement is driven by the locally enhanced electromagnetic near fields of the radial BIC resonances. We further examined the SHG signal originating from the bare silicon radial BIC structures and observed only a negligible contribution to the overall SHG intensity (see Supplementary Fig. S11a).
We further conducted control experiments where we suppressed the radial BIC resonance by investigating a symmetric structure (ΔL = 0 nm, Supplementary Fig. S12). In this case, we observe no enhancement of the SHG signal when the monolayer is on top of symmetric rings and probed at the wavelength associated with the radial BIC resonance at 744 nm, further confirming that the observed SHG enhancement arises from the near-field coupling of the MoSe 2 monolayer with the radial BIC mode. We also perform linearly polarized excitation and show the corresponding polarization-dependent SHG maps in Supplementary  Fig. S13. Crucially, the SHG generation experiments are highly facilitated by the compact footprint and polarization invariance of our ring geometry since the constraints on the spatial alignment and orientation of the TMD monolayer are released.

Discussion
We have introduced a new platform for enhancing light-matter interactions that combines all benefits of symmetry-broken qBIC approaches while providing at least 50 times less footprint compared to extended 2D metasurfaces 19 . We have demonstrated the enormous potential of our nanophotonic framework by showing highly sensitive biomolecular detection experiments as well as localized SHG enhancement in a monolayer of MoSe 2 . Our results enable on-demand tailoring of high-Q resonances in ultracompact optical devices for a variety of applications such as integrated multiresonant and hybrid sensors as well as photonic modes for coupled polariton physics.

Simulations
Numerical calculations were performed in the frequency domain by obtaining full-wave solutions of Maxwell's equations using the commercially available RF module of the finite element solver COMSOL Multiphysics. Port boundary conditions were employed for the excitation of the structures and perfectly matched layers (PML) domains for the absorption of propagating waves. A dispersionless refractive index of n = 1.45 was employed for the SiO 2 substrate, while the dispersive refractive index of the amorphous silicon film, as measured via ellipsometry, was used for the resonators. The field normalization was performed considering the solution obtained for the sum of the inplane components of the electric field (E x + E y ) using as a reference the smallest chosen asymmetry (ΔL = 25 nm) to highlight the reduction in field enhancement for larger asymmetries (for near-field intensities, see Supplementary Fig. S6).

Radial BIC fabrication
Prior to fabrication, the fused silica substrates were cleaned in an ultrasonic bath at 55°C with acetone and rinsed with isopropanol (IPA) followed by an oxygen plasma etching step to guarantee the cleanliness of the substrates. Afterwards, a 120 nm thick amorphous silicon (a-Si) layer was deposited at 250°C via plasma-enhanced chemical vapor deposition on top of the silica substrate. We used a 180 nm thick double layer of the positive tone resist poly (methyl methacrylate) (PMMA) with different chain lengths (495 k and 950 k) to guarantee an undercut of the resist after development which aids for the liftoff process. We baked each resist layer at 170°C for 3 min. A conductive polymer (E-Spacer 300Z) was deposited on top of the PMMA double layer to suppress charge accumulations on the substrate.
Electron beam lithography (EBL) was utilized to define the nanostructure pattern into the resist using 30 kV acceleration voltage for maximum edge sharpness of our nanostructures. The illuminated resist was developed in a 3:1 IPA:MIBK (Methyl isobutyl ketone) for 90 seconds after the organic polymer has been washed of by a water bath for 10 seconds. We used a combination of 20 nm silicon dioxide (SiO 2 ) and 25-nm chromium (Cr) layer deposited via electron beam evaporation as hard mask where the SiO 2 layer served as an effective diffusion barrier for the Cr atoms which can introduce severe losses within the silicon radial BIC structures. The liftoff of the metal structures was performed in special remover (Microsposit remover 1165) at 80°C overnight. The hard mask pattern was transferred into the silicon film via inductively coupled reactive ion etching (ICP-RIE) by chlorine and fluorine chemistry. After removing the Cr hard mask with a chromium wet etch, we removed the SiO 2 hard mask by a final ICP-RIE step to obtain the pure silicon nanostructures.

Optical measurements
The optical characterization of our radial BIC structures was conducted with a commercial white light transmission microscopy setup (WiTec alpha300 series). Importantly, the samples were illuminated with collimated white light from the backside and the transmitted signal was collected with a ×50 objective (NA = 0.8). Because of the collimated and linearly polarized illumination, good agreement could be obtained with the simulations, where the same illumination conditions were assumed. The collected light was focused into a fiber which is coupled to a grating-spectrometer equipped with a silicon CCD camera that we used to acquire white light transmittance spectra. All the transmittance spectra shown were referenced to the bare SiO 2 substrate. We note that the interaction cross-section of the white light depends on the ring radius of the constituent geometry, thus, different radii lead to different transmitted white light intensities potentially causing slightly different resonance modulations. This behavior, however, did not impede our physical observations and our conclusions are in excellent agreement with the theory.

Biofunctionalization
The radial BIC structures were treated by piranha solution and oxygen plasma cleaning processes to activate the structure surface with a high amount of hydroxyl radicals (−OH). Then, a silanization protocol using (3-aminopropyl)triethoxysilane (APTES, 10% v/v in ethanol, overnight) was used to modify the structure surface with NH 2 groups, which is ready to capture sulfo-NHS-biotin molecules (ThermoFisher Scientific, Mn = 443.4, as a linker between the radial resonators and target molecules). The sulfo-NHS-biotin molecules dissolved in phosphate buffered saline (1×, PBS) buffer solution were drop-casted on the structures with a volume of 40 µL and allowed to incubate for 2 h at room temperature, followed by a rinsing step using 1× PBS buffer and drying under N 2 flow. Afterwards, the biotin-immobilized resonators were incubated in target analyte solutions of streptavidin with varying concentrations for 2 h at room temperature. Once the streptavidin incubation finished, the resonators were rinsed with PBS buffer, dried using N 2 flow, and transferred to the optical characterization setup.

Determination of quality factors
For the Q factor extraction of our sharp radial BIC resonances, we employed temporal coupled mode theory 39 to extract the linewidth γ of the resonance (see Supplementary Fig. S9) using where e iϕ t 0 accounts for the background transmission and controls the shape of the resonance. The Q factor was calculated as 32 where λ res is the resonance wavelength and γ i and γ r is the intrinsic and radiative loss rate, respectively. The same fitting approach was applied for the simulated and experimental transmittance spectra to ensure consistent Q factors.

Extraction of bulk refractive index sensitivity
The bulk refractive index sensitivity was extracted from the surface sensitivity S S , which can be calculated using where n S is the refractive index of the conformally deposited SiO 2 thin film 34 . Similar to the bulk figure of merit for refractive index sensing, we defined the figure of merit for surface sensitivity as with FWHM being the full width at half maximum of the radial BIC resonance. From the figure of merit for surface sensitivity, we could calculate the figure of merit for bulk refractive index sensitivity via where h is the thickness of the thin films and l d is the decay length of the electric near fields that we extracted from simulations as 44.7 nm.

2D material fabrication and transfer
The exfoliation of a large area atomically thin monolayers of molybdenum diselenide (MoSe 2 ) was done via mechanical exfoliation from a seed crystal (HQ Graphene) with blue tape (Nitto) onto polydimethylsiloxane (PDMS) stamps. Suited monolayers for transfer were identified with an optical microscope, and monolayers were confirmed by photoluminescence imaging on the PDMS substrate. For the transfer, we used a home-built transfer setup with motorized z stage to control the stamp movement.

SHG measurements
We employed the output of a tunable Ti:sapphire laser (Coherent Chameleon Ultra II) to selectively excite the SHG signal from the MoSe 2 monolayer sample mounted in an inverted microscope setup and collected in back reflection geometry. The sample was placed on a motorized stage allowing the mapping of the sample surface. The SHG signal was isolated from the excitation beam with a short pass filter (Semrock) and collected with an avalanche photodiode detector (Micro Photon Devices). The polarization-resolved SHG signal was obtained by rotating the linear polarization of the excitation beam with a halfwave plate. The signal was then filtered with a linear polarizer, either parallel or perpendicular to the initial excitation laser polarization, before being collected in a grating-spectrometer equipped with a silicon CCD camera (Princeton Instruments).

Data availability
The main data supporting the findings of this study are available within the article and its Supplementary Information files. Extra data are available from the corresponding author upon reasonable request.